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Advective Time Lags in Box Models

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  • 1 Department of Physics, North Dakota State University, Fargo, North Dakota
  • | 2 Mathematics Department and Program in Applied Mathematics, The University of Arizona, Tucson, Arizona
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Abstract

A box model of the thermohaline circulation with mixed boundary conditions in which advective processes are incorporated via an explicit time delay mechanism is considered. The pipes that connect the subtropical and subpolar boxes have a finite volume and do not interact with the atmosphere or with the rest of the ocean except for channeling fluxes between the subtropical and subpolar regions. The configuration can be reduced to a two-box model, which, unlike the traditional Stommel model, incorporates finite-time advective processes. It is found that including a time lag leaves the haline dominant steady state stable, but the thermally dominant steady state, which is stable in Stommel's model, can have an oscillatory instability. However, this instability does not lead to sustained oscillations. Instead, it simply makes the circulation cross over to the stable haline dominant pattern. Even in part of the parameter range for which the thermally dominant state remains linearly stable, the time lag leads to a finite-amplitude instability so that a relatively small—but not infinitesimal—perturbation about the thermal state can switch the circulation to the haline state.

Corresponding author address: Prof. Juan M. Restrepo, Mathematics Department, Building 89, The University of Arizona, Tucson, AZ 85721. Email: restrepo@math.arizona.edu

Abstract

A box model of the thermohaline circulation with mixed boundary conditions in which advective processes are incorporated via an explicit time delay mechanism is considered. The pipes that connect the subtropical and subpolar boxes have a finite volume and do not interact with the atmosphere or with the rest of the ocean except for channeling fluxes between the subtropical and subpolar regions. The configuration can be reduced to a two-box model, which, unlike the traditional Stommel model, incorporates finite-time advective processes. It is found that including a time lag leaves the haline dominant steady state stable, but the thermally dominant steady state, which is stable in Stommel's model, can have an oscillatory instability. However, this instability does not lead to sustained oscillations. Instead, it simply makes the circulation cross over to the stable haline dominant pattern. Even in part of the parameter range for which the thermally dominant state remains linearly stable, the time lag leads to a finite-amplitude instability so that a relatively small—but not infinitesimal—perturbation about the thermal state can switch the circulation to the haline state.

Corresponding author address: Prof. Juan M. Restrepo, Mathematics Department, Building 89, The University of Arizona, Tucson, AZ 85721. Email: restrepo@math.arizona.edu

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